How to turn my honeycomb pattern into the sphere?
You may achieve it beveling the Ico Sphere (Percent mode) and inseting the faces.


Add an Ico Sphere and subdivide it (in my case I've made 3 subdivisions).


Select the whole mesh in Edit Mode with A and press Ctrl+B to bevel it. Use the Mouse Scroll to reduce segments to 1. Press M twice to enable the Percent mode and scale hexagons to lay as close to each other as possible. Next remove doubles with W-->Remove Doubles. Increase the Merge Distance value if needed.

NOTE: You can achieve similar effect using the Bevel Modifier, but geometry may be more dense then.


Now add a Subsurf Modifier with some subdivisions and smooth the object in the Tool Shelf's (T) Edit panel. With the whole mesh selected press I twice to inset individual faces, then scale the faces inwards to create the holes.


Finally with the whole mesh selected (A) press Shift+E to crease the edges. Play with the Mean Crease value in the Transform panel of Properties Shelf (N) to adjust its strenght as you like. 


FINAL NOTE: The model has pentagons as @kheetor warned about, but they're not much noticable.